Mulitgraded Dyson-Schwinger systems

TL;DR

This paper investigates the algebraic structure of systems of Dyson-Schwinger equations with multiple coupling constants, classifying when the generated subalgebras are Hopf and relating these to quantum field theories.

Contribution

It introduces a classification of pre-Lie algebras for Dyson-Schwinger systems and characterizes when the generated subalgebras are Hopf, extending the understanding of combinatorial quantum field theory structures.

Findings

Hopf subalgebras occur when the number of coupling constants matches the number of interactions in the QFT.

For QED, , and QCD, this number is minimal for the Hopf property.

Generalizations of fundamental and cyclic Dyson-Schwinger systems are provided.

Abstract

We study systems of combinatorial Dyson-Schwinger equations with an arbitrary number $N$ of coupling constants. The considered Hopf algebra of Feynman graphs is $\mathbb{N}^N$-graded, and we wonder if the graded subalgebra generated by the solution is Hopf or not. We first introduce a family of pre-Lie algebras which we classify, dually providing systems generating a Hopf subalgebra, we also describe the associated groups, as extensions of groups of formal diffeomorphisms on several variables. We then consider systems coming from Feynman graphs of a Quantum Field Theory. We show that if the number $N$ of independent coupling constants is the number of interactions of the considered QFT, then the generated subalgebra is Hopf. For QED, $\varphi^3$ and QCD, we also prove that this is the minimal value of $N$. All these examples are generalizations of the first family of Dyson-Schwinger systems in the one coupling constant case, called fundamental.We also give a generalization of the second family, called cyclic.